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R E S E A R C H

Open Access

Some results in quasi-

b

-metric-like spaces

Chuanxi Zhu, Chunfang Chen

*

and Xiaozhi Zhang

*Correspondence: ccfygd@sina.com

Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China

Abstract

In this paper, we introduce a new concept of quasi-b-metric-like spaces as a

generalization ofb-metric-like spaces and quasi-metric-like spaces. Some fixed point theorems are investigated in quasi-b-metric-like spaces. Moreover, an example is given to support one of our results.

MSC: 47H10; 54H25

Keywords: quasi-b-metric-like spaces; common fixed point; fixed point

1 Introduction and preliminaries

It is well known that the theoretical framework of metric fixed point theory has been an active research field and the contraction mapping principle is one of the most im-portant theorems in functional analysis. Many authors have devoted their attention to generalizing metric spaces and the contraction mapping principle. In [, ], Matthews in-troduced the notion of partial metric space as a part of the study of denotational seman-tics of dataflow networks. The partial metric space is a generalization of the metric space. Many other generalized metric spaces, such asb-metric spaces [], partialb-metric spaces [], quasi-partial metric spaces [], dislocated metric spaces [] andb-dislocated metric spaces [], were introduced. Fixed point theorems were studied in the above generalized metric spaces (see,e.g., [–] and the references therein).

The notion of metric-like spaces was introduced by Amini-Harandi in [].

Definition .[] A mappingσ:X×X→[, +∞), whereXis a nonempty set, is said to be metric-like onXif for anyx,y,zX, the following three conditions hold true:

(σ) σ(x,y) = ⇒x=y;

(σ) σ(x,y) =σ(y,x);

(σ) σ(x,z)≤σ(x,y) +σ(y,z).

The pair (x,σ) is then called a metric-like space.

Recently, the concept ofb-like spaces, which is a new generalization of metric-like spaces and partial metric spaces, was introduced by Alghamdiet al.[].

Definition . [] Ab-metric-like on a nonempty set X is a function D:X×X

[, +∞) such that for allx,y,zXand a constant s≥, the following three conditions hold true:

(D) ifD(x,y) = thenx=y;

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(D) D(x,y) =D(y,x);

(D) D(x,z)≤s[D(x,y) +D(y,z)].

The pair (x,D) is then called ab-metric-like space.

In [], some concepts inb-metric-like spaces were introduced as follows.

Eachb-metric-likeDonXgeneralizes a topologyτD onXwhose base is the family of

openD-ballsBD(x,ε) ={yX:|D(x,y) –D(x,x)|<ε}for allxXandε> .

A sequence{xn}in theb-metric-like space (X,D) converges to a pointxXif and only

ifD(x,x) =limn→+∞D(x,xn).

A sequence{xn}in theb-metric-like space (X,D) is called a Cauchy sequence if there

exists (and is finite)limn,m→+∞D(xm,xn).

A b-metric-like space is called to be complete if every Cauchy sequence {xn} in X

converges with respect to τD to a point xX such that limn→+∞D(x,xn) =D(x,x) =

limn,m→+∞D(xm,xn).

In [], Zhuet al.introduced the concept of quasi-metric-like spaces and investigated some fixed point theorems in quasi-metric-like spaces.

Definition .[] LetXbe a nonempty set. A mappingρ:X×X→[, +∞) is said to be a quasi-metric-like onXif for anyx,y,zXthe following conditions hold:

() ρ(x,y) = ⇒x=y; () ρ(x,z)≤ρ(x,y) +ρ(y,z).

The pair (x,ρ) is then called a quasi-metric-like space.

In this paper, inspired by Definitions . and ., we define a quasi-b-metric-like which generalizes the quasi-metric-like and b-metric-like. Furthermore, we investigate some fixed point theorems in quasi-b-metric-like spaces. Also, we give an example to illustrate the usability of one of the obtained results.

2 Main results

In this section, we begin with introducing the concept of a quasi-b-metric-like space.

Definition . A quasi-b-metric-like on a nonempty set X is a functionb:X×X

[, +∞) such that for all x,y,zXand a constants≥, the following conditions hold true:

(qb) b(x,y) = ⇒x=y; (qb) b(x,z)≤s[b(x,y) +b(y,z)].

The pair (X,b) is then called a quasi-b-metric-like space. The numbersis called to be the coefficient of (X,b).

Example . LetX={, , }, and let

b(x,y) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

, x=y= , 

, x= ,y= , , x= ,y= ,

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Then (X,b) is a quasi-b-metric-like space with the coefficient s= , but sinceb(, )=

b(, ), then (X,b) is not ab-metric-like space. It is obvious that (X,b) is not a quasi-metric-like space.

Definition . Let (X,b) be a quasi-b-metric-like space. Then

() A sequence{xn}in(X,b)converges to a pointxXif and only if

lim

n→+∞b(xn,x) =n→+∞lim b(x,xn) =b(x,x).

() A sequence{xn}in(X,b)is called a Cauchy sequence iflimn,m→+∞b(xn,xm)and

limn,m→+∞b(xm,xn)exist and are finite.

() A quasi-b-metric-like space(X,b)is called to be complete if for every Cauchy sequence{xn}in(X,b), there exists somexXsuch that

lim

n→+∞b(xn,x) =n→+∞lim b(x,xn) =b(x,x) =n,mlim→+∞b(xn,xm) =n,mlim→+∞b(xm,xn).

() A sequence{xn}in(X,b)is called a -Cauchy sequence if

lim

n,m→+∞b(xn,xm) =n,mlim→+∞b(xm,xn) = .

() A quasi-b-metric-like space(X,b)is called to be -complete if for every -Cauchy sequence{xn}inX, there exists somexXsuch that

lim

n→+∞b(xn,x) =n→+∞lim b(x,xn) =b(x,x) =  =n,mlim→+∞b(xn,xm) =n,mlim→+∞b(xm,xn).

It is obvious that every -Cauchy sequence is a Cauchy sequence in the quasi-b -metric-like space (X,b), and every complete quasi-b-metric-like space is a -complete quasi-b -metric-like space, but the converse assertions of these facts may not be true.

Remark . In Example ., letxn=  forn= , , . . . , then it is clear thatlimn→+∞b(xn, ) =

limn→+∞b(,xn) =b(, ) andlimn→+∞b(xn, ) =limn→+∞b(,xn) =b(, ). Therefore, in

quasi-b-metric-like spaces, the limit of a convergent sequence is not necessarily unique.

Now we prove our main results.

Theorem . Let(X,b)be a-complete quasi-b-metric-like space with the coefficient s≥,

and let f :X×XX be a mapping such that

bf(x),f(y)≤ϕb(x,y) (.)

for all x,yX,whereϕ: [, +∞)→[, +∞)is a continuous mapping such thatϕ(t) = if and only if t= andϕ(t) <t for all t> .Ifn=snϕn(t)converges for all t> ,whereϕnis

the nth iterate ofϕ,then f has a unique fixed point.Moreover,for any x∈X,the iterative

sequence{fn(x)}converges to the fixed point.

Proof Letxbe an arbitrary point inX. From (.), we have

bfn(x),fn+(x)

ϕbfn–(x),fn(x)

≤ · · · ≤ϕnbx,f(x)

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and

bfn+(x),fn(x)

ϕbfn(x),fn–(x)

≤ · · · ≤ϕnbf(x),x

, n> . (.)

Ifb(x,f(x)) =  orb(f(x),x) = , thenx=f(x), which means thatxis a fixed point off. Suppose thatb(x,f(x)) >  andb(f(x),x) > . Now we show that{fn(x)}is a -Cauchy sequence. For any integerrZ+(the set of positive integers), the property (qb) implies that

bfn(x),fn+r(x)

s bfn(x),fn+(x)

+bfn+(x),fn+r(x)

sbfn(x),fn+(x)

+sbfn+(x),fn+(x)

+bfn+(x),fn+r(x)

sbfn(x),fn+(x)

+sbfn+(x),fn+(x)

+sbfn+(x),fn+(x)

+bfn+(x),fn+r(x)

.. .

sbfn(x),fn+(x)

+sbfn+(x),fn+(x)

+sbfn+(x),fn+(x)

+· · ·

+sr–bfn+r–(x),fn+r–(x)

+sr–bfn+r–(x),fn+r(x)

sbfn(x),fn+(x)

+sbfn+(x),fn+(x)

+sbfn+(x),fn+(x)

+· · ·

+sr–bfn+r–(x),fn+r–(x)

+srbfn+r–(x),fn+r(x)

. (.)

Equations (.) and (.) yield that

bfn(x),fn+r(x)

sϕnbx,f(x)

+sϕn+bx,f(x)

+sϕn+bx,f(x)

+· · ·

+sr–ϕn+r–bx,f(x)

+srϕn+r–bx,f(x)

snϕnbx,f(x)

+sn+ϕn+bx,f(x)

+sn+ϕn+bx,f(x)

+· · ·

+sn+r–ϕn+r–bx,f(x)

+sn+r–ϕn+r–bx,f(x)

=

n+r–

k=n

skϕkbx,f(x)

. (.)

Since∞n=snϕn(t) converges for allt> , thenlim

n→+∞b(fn(x),fn+r(x)) = , which means that form>n,

lim

n,m→+∞b

fn(x),fm(x)

= . (.)

Also, applying (.), we proceed similarly as above and obtain limn→+∞b(fn+r(x

),

fn(x)) = , which means that form>n,

lim

n,m→+∞b

fm(x),fn(x)

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From (.) and (.), we get that {fn(x

)} is a -Cauchy sequence. Since (X,b) is -complete, then the sequence{fn(x

)}converges to some pointzX, that is,

lim

n→+∞b

fn(x),z

= lim

n→+∞b

z,fn(x)

=b(z,z) =  = lim

n,m→+∞b

fn(x),fm(x)

= lim

n,m→+∞b

fm(x),fn(x)

. (.)

We now show thatzis a fixed point off. By the triangle inequality, we have

b(z,fz)≤s bz,fn+(x)

+bfn+(x),fz

=s bz,fn+(x)

+bffn(x)

,fz

sbz,fn+(x)

+sϕbfn(x),z

.

Using (.) in the above inequalities, we obtainb(z,fz) = , that is,fz=z, hencezis a fixed point off. Next, we show thatzis the unique fixed point off. Suppose thatuis also a fixed point off, then we claimb(z,u) = . Suppose that this is not the case, then

b(z,u) =b(fz,fu)≤ϕb(z,u)<b(z,u).

It is a contradiction, henceb(z,u) = , which impliesz=u, thereforef has a unique fixed

point.

In Theorem ., takingϕ(t) =λtwith ≤λ<s, we can get the following corollary.

Corollary . Let (X,b)be a-complete quasi-b-metric-like space with the coefficient s≥,and let f:XX be a mapping such that

b(fx,fy)≤λb(x,y) (.)

for all x,yX,where≤λ<s.Then f has a unique fixed point in X.Moreover,for any x∈X,the iterative sequence{fn(x)}converges to the fixed point.

Theorem . Let (X,b)be a-complete quasi-b-metric-like space with the coefficient s≥,and let F:X×XX be a mapping.If there exists k∈[,s)such that

bF(x,y),F(u,v)≤k

b(x,u) +b(y,v)

(.)

for each(x,y), (u,v)∈X×X,then F has a coupled fixed point,that is,there exists(x,y)such that x=F(x,y)and y=F(y,x).

Proof LetM=X×Xand define

ρ(x,y), (x,y)

=b(x,x) +b(y,y)

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u= (u,u),v= (v,v). From (.), we haveρ(Tu,Tv)≤(u,v). Applying Corollary ., we obtain thatT has a unique fixed point inX×X, hence there exists a unique (x,y)∈

X×Xsuch thatT(x,y) = (x,y), that is, (F(x,y),F(y,x)) = (x,y). Therefore,F(x,y) =xand

F(y,x) =y, which implies thatFhas a unique coupled fixed point.

Lemma .[] Let X be a nonempty set and T:XX be a mapping.Then there exists a subset EX such that T(E) =T(X)and T:EX is one-to-one.

The following definitions can be seen in [–].

Definition . Letf andgbe two self-mappings on a setX. Ifω=fx=gxfor somexinX,

thenxis called a coincidence point off andg, whereωis called a point of coincidence of

f andg.

Definition . Letf andgbe two self-mappings defined on a setX. Then f andg are

said to be weakly compatible if they commute at every coincidence point,i.e., iffx=gxfor somexX, thenfgx=gfx.

Theorem . Let(X,b)be a quasi-b-metric-like space with the coefficient s≥,and let f,

g be self-mappings on X which satisfy the following condition:

b(fx,fy)≤λb(gx,gy) (.)

for all x,yX,where≤λ<

s.If f(X)⊆g(X)and g(X)is a-complete subset of X,then f

and g have a unique point of coincidence in X.Moreover,if f and g are weakly compatible,

then f and g have a unique common fixed point.

Proof By Lemma ., there existsEXsuch thatg(E) =g(X) andg:EXis one-to-one. Now, define a mappingh:g(E)→g(E) byh(gx) =fx. Sincegis one-to-one onE,h

is well defined. Note thatb(h(gx),h(g(y)))≤λb(gx,gy) for allg(x),g(y)∈g(E), where ≤

λ<s. Sinceg(E) =g(X) is -complete, by using Corollary ., there exists a uniquex∈X such that h(gx) =gx, hencefx=gx, which means thatf andg have a unique point of coincidence inX. Letfx=gx=z, sincef andg are weakly compatible, thenfz=gz, which together with the uniqueness of the point of coincidence implies thatz=fz=gz.

Therefore,zis the unique common fixed point off andg.

Now, we give an example to illustrate the validity of one of our main results.

Example . LetX={, , }. Defineb:X×X→[, +∞) as follows:

b(, ) = , b(, ) = , b(, ) = ;

b(, ) = , b(, ) = , b(, ) = ;

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Then (X,b) is a complete quasi-b-metric-like space with the coefficients=. Define the mappingf:XXby

f = , f = , f = .

It is easy to prove thatf satisfies all the conditions of Corollary . withλ∈[,). Now, by Corollary .,f has a unique fixed point. In fact,  is the unique fixed point off.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the work. All authors read and approved the final manuscript.

Acknowledgements

The authors are thankful to the referees for their valuable comments and suggestions to improve this paper. The research was supported by the National Natural Science Foundation of China (11361042, 11071108) and supported by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201007, 20142BAB201007, 20142BAB211004) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ13081).

Received: 30 June 2014 Accepted: 14 October 2014 Published:31 Oct 2014 References

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Cite this article as:Zhu et al.:Some results in quasi-b-metric-like spaces.Journal of Inequalities and Applications

References

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